If something like a (pan-)Slavic organization really had to exist, that would cover all these things (festivals, games, celebrations, holidays and other events), its members would have to speak a common language. There are tons of abandoned projects of inter-Slavic languages that didn’t make it. I know that today Slavs are able to communicate in English very well with each other, but I would like to see one of those projects to be used instead.
For example: http://steen.free.fr/interslavic/index.html
All Slavs should understand most of it without learning, so learning to speak it shouldn’t be a problem.